Kybernetika 54 no. 1, 41-60, 2018

On the static output feedback stabilization of deterministic finite automata based upon the approach of semi-tensor product of matrix

Zhipeng Zhang, Zengqiang Chen, Xiaoguang Han and Zhongxin LiuDOI: 10.14736/kyb-2018-1-0041


In this paper, the static output feedback stabilization (SOFS) of deterministic finite automata (DFA) via the semi-tensor product (STP) of matrices is investigated. Firstly, the matrix expression of Moore-type automata is presented by using STP. Here the concept of the set of output feedback feasible events (OFFE) is introduced and expressed in the vector form, and the stabilization of DFA is defined in the sense of static output feedback (SOF) control. Secondly, SOFS problem of DFA is investigated within the framework of STP, including single-equilibrium-based SOFS, multi-equilibrium-based SOFS, and further limit cycle-based SOFS. Then the necessary and sufficient conditions for the existence of the three types SOFS are proposed respectively. Meanwhile the efficient and systematic procedures based on the matrix theory to seek the corresponding SOF controller are provided for the three types SOFS problem. Finally, two examples are presented to illustrate the effectiveness of the proposed approach.


finite automata, discrete event dynamic systems, static output feedback stabilization, semi-tensor product, output feedback feasible events


93D15, 93C65


  1. N. Bof, E. Fornasini and M. Valcher: Output feedback stabilization of Boolean control networks. Automatica 57 (2015), 21-28.   DOI:10.1016/j.automatica.2015.03.032
  2. C. Cassandras and S. Lafortune: Introduction to Discrete Event System. Second edition. Springer Science and Business Media, New York 2008.   DOI:10.1007/978-0-387-68612-7
  3. D. Cheng: Disturbance decoupling of Boolean control networks. IEEE Trans. Automat. Control 56 (2011), 2-10.   DOI:10.1109/tac.2010.2050161
  4. D. Cheng, F. He, H. Qi and T. Xu: Modeling, analysis and control of networked evolutionary games. IEEE Trans. Automat. Control 60 (2015), 2402-2415.   DOI:10.1109/tac.2015.2404471
  5. D. Cheng and H. Qi: Controllability and observability of Boolean control networks. Automatica 45 (2009), 1659-1667.   DOI:10.1016/j.automatica.2009.03.006
  6. D. Cheng and H. Qi: A linear representation of dynamics of Boolean networks. IEEE Trans. Automat. Control 55 (2010), 2251-2258.   DOI:10.1109/tac.2010.2043294
  7. R. Daniel and L. Markus: Automata with modulo counters and nondeterministic counter bounds. Kybernetika 50 (2014), 66-94.   DOI:10.14736/kyb-2014-1-0066
  8. E. Fornasini and M. E Valcher: On the periodic trajectories of Boolean control networks. Automatica 49 (2013), 1506-1509.   DOI:10.1016/j.automatica.2013.02.027
  9. J. Holub: The finite automata approaches in stringology. Kybernetika 48 (2012), 386-401.   CrossRef
  10. X. Han, Z. Chen, Z. Liu and et al.: Calculation of siphons and minimal siphons in Petri nets based on semi-tensor product of matrices. IEEE Trans. Systems, Man Cybernet.: Systems 47 (2017), 531-536.   DOI:10.1109/tsmc.2015.2507162
  11. X. Han, Z Chen, Z. Liu and et al.: The detection and stabilisation of limit cycle for deterministic finite automata. Int. Control 91 (2017), 4, 874-886.   DOI:10.1080/00207179.2017.1295319
  12. A. Kobetski and M Fabian: Time-optimal coordination of flexible manufacturing systems using deterministic finite automata and mixed integer linear programming. Discrete Event Dynamic Systems 19 (2009), 287-315.   DOI:10.1007/s10626-009-0064-9
  13. H. Li and Y. Wang: Output feedback stabilization control design for BCNs. Automatica 49 (2013), 3641-3645.   DOI:10.1016/j.automatica.2013.09.023
  14. Z. Li, Y. Qiao, H. Qi and D. Cheng: Stability of switched polynomial systems. J. Systems Science Complexity 21 (2008), 362-377.   DOI:10.1007/s11424-008-9119-5
  15. C. Ozveren and A. Willsky: Output stabilizability of discrete event dynamic systems. IEEE Trans. Automat. Control 19 (1991), 925-935.   DOI:10.1109/9.133186
  16. K. Passino, A. Michel and P. Antsaklis: Lyapunov stability of a class of discrete event systems. IEEE Trans. Automat. Control 39 (1994), 269-279.   DOI:10.1109/9.272323
  17. V. Syrmos, C. Abdallah, P. Dorato and et al.: Static output feedback - A survey. Automatica 33 (1997), 125-137.   DOI:10.1016/s0005-1098(96)00141-0
  18. S. P. Tiwari and A. K. Srivastava: On a decomposition of fuzzy automata. Fuzzy Sets Systems 151 (2005), 503-511.   DOI:10.1016/j.fss.2004.06.014
  19. X. Xu and Y. Hong: Matrix expression and reachability analysis of finite automata. J. Control Theory Appl. 10 (2012), 210-215.   DOI:10.1007/s11768-012-1178-4
  20. X. Xu, Y. Zhang and Y. Hong: Matrix approach to stabilization of deterministic finite automata. In: Proc. American Control Conference, Washington 2013, pp. 3242-3247.   DOI:10.1109/acc.2013.6580331
  21. Y. Yan, Z. Chen and Z. Liu: Solving type-2 fuzzy relation equations via semi-tensor product of matrices. Control Theory Technol. 12 (2014), 173-186.   DOI:10.1007/s11768-014-0137-7